Non-autonomous right and left multiplicative perturbations and maximal regularity
Volume 242 / 2018
Studia Mathematica 242 (2018), 1-29
MSC: Primary 35K90, 35K50; Secondary 47D06.
DOI: 10.4064/sm8721-6-2017
Published online: 11 January 2018
Abstract
We consider the problem of maximal regularity for non-autonomous Cauchy problems $$ u’(t) + B(t) A(t) u(t) + P(t) u(t) = f(t), \ \hskip 1em u(0) = u_0, $$ and $$ u’(t) + A(t) B(t) u(t) + P(t) u(t) = f(t), \ \hskip 1em u(0) = u_0. $$ In both cases, the time dependent operators $A(t)$ are associated with a family of sesquilinear forms, and the multiplicative left or right perturbations $B(t)$ as well as the additive perturbation $P(t)$ are families of bounded operators on the Hilbert space considered. We prove maximal $L_p$-regularity results and other regularity properties for the solutions of the previous problems under minimal regularity assumptions on the forms and perturbations.
